Muon conversion to electron in nuclei in Minimal R-symmetric Supersymmetric Standard Model

We analyze the lepton flavor violating process $\mu-e$ conversion in the framework of the minimal R-symmetric supersymmetric standard model. The theoretical predictions are determined by considering the experimental constraint on parameter $\delta^{12}$ from the lepton flavor violating decay $\mu\rightarrow e \gamma$. The numerical results show that $\gamma$ penguins and Z penguins dominate the predictions on CR($\mu-e$,Nucleus), and the contributions from Higgs penguins and box diagrams are insignificant. The theoretical predictions on conversion rate CR($\mu-e$,Nucleus) in a Al or Ti target can be enhanced close to the future experimental sensitivities and are very promising to be observed in near future experiment.


I. INTRODUCTION
Searching for Lepton Flavor Violating (LFV) decays are of great importance in probing New Physics (NP) beyond the Standard Model (SM) in which the theoretical predictions on those LFV decays are suppressed by small masses of neutrinos and far beyond the experimental accessibility. There are many different ways to search LFV such as µ → eγ, µ → 3e, µ − e conversion in nucleus, τ decays, hadron decays and so on. However, no LFV signals have been observed in experiment up to now. The µ − e conversion in nucleus is a process that muons are captured in a target of atomic nucleus and form a muonic atom. Several experiments have been built or planned to built to search for this process. Current limit on the µ − e conversion rate is 4.6 × 10 −12 for a Ti target at TRIUMF [1], 4.3 × 10 −12 for a Ti target and 7 × 10 −13 for a Au target at SINDRUM-II experiment [2]. In future, this LFV process may be observed by experiments with improved sensitivity. A future prospects of 10 −13 for a C target or 10 −14 for a SiC target at DeeMe [3], 10 −18 for a Ti target at PRISM [4] and 10 −16 − 10 −17 for a Al target at Mu2e and COMET [5,6] will be achieved, which improve the current experimental limits by several orders of magnitude.
In this paper, we will study the LFV process µ−e conversion in the Minimal R-symmetric Supersymmetric Standard Model (MRSSM) [35]. The MRSSM has an unbroken global U(1) R symmetry and provides a new solution to the supersymmetric flavor problem in MSSM. In this model, R-symmetry forbids Majorana gaugino masses, µ term, A terms and all left-right squark and slepton mass mixings. The R-charged Higgs SU(2) L doubletsR u andR d are introduced in MRSSM to yield the Dirac mass terms of higgsinos. Additional superfieldsŜ,T andÔ are introduced to yield Dirac mass terms of gauginos. Studies on phenomenology in MRSSM can be found in literatures [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. Similar to MSSM, the off-diagonal entries δ ij in slepton mass matrices m 2 l and m 2 r dominate the LFV process µ − e conversion. Taking account of the constraints from radiative decays µ → eγ on the off-diagonal parameters δ ij , we explore µ − e conversion rate as a function of off-diagonal parameter δ ij and other model parameters.
The paper is organized as follows. In Section II, we present the details of the MRSSM. All relevant mass matrices and mixing matrices are provided. Feynman diagrams contributing to µ − e conversion in MRSSM are given at one loop level. The µ − e conversion rate are computed in effective Lagrangian method, and notations and conventions for effective operators and Wilson coefficients are also listed. The numerical results are presented in Section III, and the conclusion is drawn in Section IV.

II. MRSSM
In this section, we firstly provide a simple overview of MRSSM in order to fix the notations we use in this paper. The MRSSM has the same gauge symmetry SU(3) C × SU(2) L × U(1) Y as the SM and MSSM. The spectrum of fields in MRSSM contains the standard MSSM matter, Higgs and gauge superfields augmented by chiral adjointsÔ,T ,Ŝ and two R-Higgs iso-doublets. The general form of the superpotential of the MRSSM is given by [36] All trilinear scalar couplings involving Higgs bosons to squarks and sleptons are forbidden in Eq.(2) cause the sfermions have an R-charge and these terms are non R-invariant, and this relaxes the flavor problem of the MSSM [35]. The Dirac nature is a manifest feature of MRSSM fermions and the soft-breaking Dirac mass terms of the singletŜ, tripletT and octetÔ take the form as whereB,W andg are usually MSSM Weyl fermions. R-Higgs bosons do not develop vacuum expectation values since they carry R-charge 2. After electroweak symmetry breaking the singlet and triplet vacuum expectation values effectively modify the µ u and µ d , and the modified µ i parameters are given by The v T and v S are vacuum expectation values ofT andŜ which carry R-charge zero.
In the weak basis (σ d , σ u , σ S , σ T ), the pseudo-scalar Higgs boson mass matrix and the diagonalization procedure are In the weak basis (φ d , φ u , φ S , φ T ), the scalar Higgs boson mass matrix and the diagonalization procedure are where the submatrices (c β = cosβ, s β = sinβ) are The number of neutralino degrees of freedom in MRSSM is doubled compared to MSSM as the neutralinos are Dirac-type. In the weak basis of four neutral electroweak two-component , the neutralino mass matrix and the diagonalization procedure are The mass eigenstates κ i and ϕ i , and physical four-component Dirac neutralinos are The number of chargino degrees of freedom in MRSSM is also doubled compared to MSSM and these charginos can be grouped to two separated chargino sectors according to their R-charge. The χ ± -charginos sector has R-charge 1 electric charge; the ρ-charginos sector has R-charge -1 electric charge. In the basis , the χ ± -charginos mass matrix and the diagonalization procedure are The mass eigenstates λ ± i and physical four-component Dirac charginos are Here, we don't discuss the ρ-charginos sector in detail since it doesn't contribute to µ − e conversion. More information about the ρ-charginos can be found in Ref. [38,40,42,52].
In MRSSM the LFV decays mainly originate from the potential misalignment in sleptons mass matrices. In the gauge eigenstate basisν iL , the sneutrino mass matrix and the diagonalization procedure are where the last two terms in mass matrix are newly introduced by MRSSM. The slepton mass matrix and the diagonalization procedure are The sources of LFV are the off-diagonal entries of the 3 × 3 soft supersymmetry breaking matrices m 2 l and m 2 r in Eqs. (8,9). From Eq. (9) we can see that the left-right slepton mass mixing is absent in MRSSM, whereas the A terms are present in MSSM.
The mass matrix for up squarks and down squarks, and the relevant diagonalization procedure are The MRSSM has been implemented in the Mathematica package SARAH [55][56][57], and we use the Feynman rules generated with SARAH-4.14.
The conversion rate CR(µ − e, Nucleus) in nuclei can be calculated by Here p e and E e (∼ m µ in the numerical evaluation) are the momentum and energy of the electron. G F and α are the Fermi constant and the fine structure constant, respectively.
Z ef f is the effective atomic charge. Z and N are the number of protons and neutrons in the nucleus. F p is the nuclear form factor and Γ capt is the total muon capture rate. The values of Z e f f , F p and Γ capt that will be used in the phenomenological analysis below are given in Table. I. At quark level, the g (i) XK factors (with i=0,1, X=L,R and K=S,V) can be written as combinations of effective couplings

=1. The g XK(q) coefficients can be written as combinations of Wilson coefficients
where Q q are the electric charge of quarks, C SLL llqq equals B K XY (C K XY ) for d-quarks (u-quarks), g RV (q) = g LV (q) |L → R and g RS(q) = g LS(q) |L → R.
In following numerical analysis, the values in Eq.(13) will be used for all results. Note that, the off-diagonal entries of squark mass matrices m 2 q , m 2 u , m 2 d and slepton mass matrices m 2 l , m 2 r in Eq.(13) are zero, i.e., the flavour mixing of squark and slepton is absent. Similarly to most supersymmetry models, the LFV processes in MRSSM originate from the off-diagonal entries of the soft breaking terms m 2 l and m 2 r , which are parameterized by mass insertion where I, J = 1, 2, 3. To decrease the number of free parameters involved in our calculation, we assume that the off-diagonal entries of m 2 l and m 2 r in Eq. (14) are equal, i.e., δ IJ l = δ IJ r = δ IJ . The experimental limits on LFV decays, such as radiative two body decays l 2 → l 1 γ, leptonic three body decays l 2 → 3l 1 , can give strong constraints on the parameters δ IJ .
In the following, we will use LFV decays µ → eγ to constrain the parameters δ 12 which has been discussed in Ref. [53]. It is noted that δ 23 and δ 13 have been set zero in following discussion since they have no effect on the predictions of CR(µ − e,Nucleus). Current limits of LFV decays µ → eγ is BR(µ → eγ) < 4.2 × 10 −13 from MEG [61] and new sensitivity for this decay channel in the future projects will be BR(µ → eγ) ∼ 6 × 10 −14 from MEG II [62]. In FIG.2 the predictions for BR(µ → eγ) and CR(µ − e,Nucleus) for Al, Ti, Sr, Sb, Au, and Pb are shown as a function of mass insertion parameter δ 12 . The prediction for BR(µ → eγ) exceeds the future experiment sensitivity at δ 12 ∼ 0.001. In a recent Ref. [53] the analytical computation and discussion of BR(µ → eγ) in MRSSM has been performed.
The valid region for δ 12 in Ref. [53] calculated with the Mathematica package Package-X is compatible with that in this work calculated with SARAH and SPheno. We clearly see that both the predictions for BR(µ → eγ) and CR(µ − e,Nucleus) in nuclei are sensitive to δ 12 , and they increase along with the increase of δ 12 which have a same behavior as those in most SUSY models(e.g. [63]). At δ 12 ∼0.001, the prediction on BR(µ → eγ) is very close to the current experimental limit, and the predictions on CR(µ − e,Nucleus) are around 10 −15 − 10 −16 which are two orders of magnitude below current experimental limits. The predicted CR(µ − e,Nucleus) for Ti is around 10 −15 and this is three orders of magnitude above future experimental sensitivity [4]. The predicted CR(µ − e,Nucleus) for Al is around 10 −16 and this is in region of the future experimental sensitivity [5,6]. and this is still two orders of magnitude above future experimental sensitivity [4]. At m l = 5 TeV, the predicted CR(µ − e,Nucleus) for Al is below 10 −16 and this is still in region of the future experimental sensitivity [5,6]. in nuclei are not sensitive to tan β, and they take values along a narrow band. This is a striking difference to MSSM [42]. Due to the existence of the transition from d-Higgsino to u-Higgsino in MSSM, which is governed by µ-term, the well-known tanβ-enhancement is   33 . We clearly see that the predictions for BR(µ → eγ) is not sensitive to M QU D , and it takes values along a narrow band. This is because there is no squark medicated diagram contributing to µ → eγ. Only the box diagrams contributing to CR(µ − e,Nucleus) depend on the squark masses. The predictions for CR(µ − e,Nucleus) show a weak dependence on M QU D , and they decrease slowly along with the increase of M QU D which have a baseline behaviour as those in Ref. [42].
We are also interesting to the effects from other parameters on the predictions of CR(µ − e,Nucleus) in MRSSM. By scanning over these parameters, such as B µ , M B D , λ d , λ u , Λ d , Λ u ,µ d and µ u , it shows the predictions for CR(µ − e,Nucleus) take values along a narrow band in valid regions.

IV. CONCLUSIONS
In this work, taking account of the constraints from µ → eγ on the parameter space, we analyze the LFV process CR(µ − e,Nucleus) in the framework of the Minimal R-symmetric Supersymmetric Standard Model.
In MRSSM, the theoretical predictions on CR(µ − e,Nucleus) mainly depend on the mass insertion δ 12 . The predictions on CR(µ − e,Nucleus) would be zero if δ 12 =0 is assumed.
Taking account of experimental bounds on radiative decays µ → eγ, the values of δ 12 is